Despite the 100 year aviation history, evidence indicates that when faced with uncontrolled roll, pitch, or yaw, pilots sometimes have difficulties in quickly responding to the situation which needs rapid action to correct in order to avoid crash. Trainings for avoiding uncontrolled roll, pitch, or yaw is either not effective enough or not correct at all. The reason for such awkward situation in the industry is that the mechanism of the uncontrolled roll, pitch, or yaw has not been understood. The relationship between the current flight simulator fidelities and real aircrafts is susceptible when the roll, pitch, and yaw motions become large enough since the current flight dynamics are based on the linearization of roll, pitch, and yaw motions, meaning that the aircraft motions have to be small enough to be accurate. In real world, however, aircraft could roll 360° in the sky, like what happened to TWA Flight 841 in 1979. Many mysterious aircraft crashes were due to loss of control caused by the nonlinear instability, a new scientific discovery made by the inventor in the book “Nonlinear Instability and Inertial Coupling Effect—The Root Causes Leading to Aircraft Crashes, Land Vehicle Rollovers, and Ship Capsizes” (ISBN 9781732632301, to be published in November 2018). To name a few, the following incidents and accidents were caused by the nonlinear instability and analyzed in the book.
The incident of TWA Flight 841 Boeing 727-31 in 1979,
the crash of Japan Airlines Flight 123 Boeing 747-100SR in 1985,
the crash of Northwest Flight 255 MD DC-9-82 in 1987,
the crash of Delta Airlines Flight 1141 Boeing 727-232 in 1988,
the crash of United Airlines Flight 585 Boeing 737-200 in 1991,
the crash of USAir Flight 405 Fokker F-28 in 1992,
the crash of B-52H strategic bomber in 1994,
the crash of USAir Flight 427 Boeing 737-300 in 1994,
the incident of Boeing 737-236 Advanced G-BGJI in 1995,
the crash of SilkAir Flight 185 Boeing 737-300 in 1997,
the crash of EgyptAir Flight 990 Boeing 767-366ER in 1999,
the crash of American Airlines Flight 587 Airbus A300-605R in 2001,
the crash of PT. Mandala Airlines Flight 091 Boeing 737-200 in 2005,
the crash of Spanair Flight 5022 MD DC-9-82 in 2008,
the crash of Air France Flight 447 Airbus A330 in 2009,
the crash of Colgan Air Flight 3407 Bombardier DHC-8-400 in 2009,
the crash of Air Algeria Flight 5017 MD-83 in 2014,
the crash of FlyDubai Flight 981 Boeing 737-800 in 2016.
A fundamental mistake has been made in dealing with the aircraft dynamics in the current academic and industry practices. For an aircraft, the governing equations for its rotational motions (roll, pitch, and yaw) are given by Math. 1 in the vector form. They were obtained based on Newton's second law of motions in the body-fixed reference frame,d{right arrow over (H)}/dt=−{right arrow over (ω)}×{right arrow over (H)}+{right arrow over (M)},   Math. 1wherein {right arrow over (ω)}=(p,q,r)=({dot over (φ)}, {dot over (θ)}, {dot over (ψ)}): the angular velocities of the vehicle; φ, θ, ψ: the roll, pitch, and yaw angle about the principal axes of inertias X, Y, Z, respectively; {right arrow over (H)}=(Ixp, Iyq, Izr): the angular momentum of the vehicle; Ix, Iy, Iz: the moment of inertias about the principal axes of inertias X, Y, Z, respectively (These parameters are constants in this frame); {right arrow over (M)}=(Mx, My, Mz): the external moments acting on the aircraft about the principal axes of inertia. In both the aviation academy and industry, the current practice to deal with Math. 1 is to make a linearization approximation first and then solve the equations because the nonlinear term −{right arrow over (ω)}×{right arrow over (H)} is too difficult to deal with. The linearization approximation makes the nonlinear term −{right arrow over (ω)}×{right arrow over (H)} disappear, and the equations becomed{right arrow over (H)}/dt={right arrow over (M)}.   Math. 2However, the equations are still considered in the body-fixed reference frame which is a non-inertial frame. The reason for this is that the external moments (Mx, My, Mz) acting on vehicles and the moments of inertia Ix, Iy, Iz are needed to be considered in the body-fixed reference frame.
The fundamental mistake is that the nonlinear term −{right arrow over (ω)}×{right arrow over (H)} cannot be neglected because they are the inertial moments tied to the non-inertial reference frame which is the body-fixed reference frame in this case. This mistake is similarly like we neglect the Coriolis force which equals −2{right arrow over (Ω)}×{right arrow over (V)}, where {right arrow over (Ω)} is the angular velocity vector of the earth and {right arrow over (V)} is the velocity vector of a moving body on earth. Then we try to explain the swirling water draining phenomenon in a bathtub. In this case, we are considering the water moving in the body-fixed and non-inertial reference frame which is the earth. The Coriolis force is an inertial force generated by the rotating earth on the moving objects which are the water particles in this case. Without the Coriolis force, we cannot explain the motions of the swirling water. Similarly in the aircraft dynamics, the aircraft is rotating, and we consider the rotational motions of the aircraft in the body-fixed and non-inertial reference frame which is the aircraft itself. The difference between the two cases is that in the former the object (water particle) has translational motions ({right arrow over (V)}) while in the latter the object (aircraft itself) has rotational motions ({right arrow over (ω)}) but they both have the important inertial effects which cannot be neglected because both the objects are considered in the non-inertial reference frames. In the former the inertial effect is the Coriolis force −2{right arrow over (Ω)}×{right arrow over (V)} while in the latter the inertial effect is the inertial moment −{right arrow over (ω)}×{right arrow over (H)} which are not forces but moments since we are dealing with rational motions instead of translational one. Without the inertial moment, we cannot explain many phenomena which happened to aircrafts, like uncommanded motions of roll, pitch, and yaw; and Pilot-Induced-Oscillation (PIO) .
In the inventor's book, the equations Math. 1 have been solved analytically without the linearization approximation and it was found that the pitch motion, without loss of generality assuming the pitch moment of inertia to be the intermediate between the roll and yaw inertias, is conditionally stable and becomes unstable in certain circumstances. A brief summary of the findings is given below. The governing equations of rotational motions of an aircraft under a periodic external pitch moment can be written in scalar form asIx{umlaut over (φ)}+b1{dot over (φ)}+k1φ=(Iy−Iz){dot over (θ)}{dot over (ψ)},   Math. 3Iy{umlaut over (θ)}+b2{dot over (θ)}+k2θ=(Iz−Ix){dot over (φ)}{dot over (ψ)}+M21 cos(ω21t+α21),    Math. 4Iz{umlaut over (ψ)}+b3{dot over (ψ)}+k3ψ=(Ix−Iy){dot over (φ)}{dot over (θ)},   Math. 5wherein b1, b2, b3 are the damping coefficients for roll, pitch, and yaw, respectively; k1, k2, k3 are the restoring coefficients for roll, pitch, and yaw, respectively; M21 is the external pitch moment amplitude; ω21 and α21 are the frequency and phase of the external pitch moment, respectively. These equations represent a dynamic system governing the rotational dynamics of an aircraft when taking off or approaching to landing. According to the current practice in the industries under the linearization approximation, these equations becomeIx{umlaut over (φ)}+b1{dot over (φ)}+k1φ=0,   Math. 6Iy{umlaut over (θ)}+b2{dot over (θ)}+k2θ=M21 cos(ω21+α21),   Math. 7Iz{umlaut over (ψ)}+b3{dot over (ψ)}+k3ω=0.   Math. 8Therefore the current practice says that the aircraft will only have pitch motion, no roll and yaw motions because there are no moments acting on roll and yaw directions. In reality, however, there exist moments acting in roll and yaw directions as indicated by the nonlinear terms in the right hand sides of Math. 3 and Math. 5, respectively. These moments are the components of the inertial moment vector −{right arrow over (ω)}×{right arrow over (H)} along roll and yaw directions, respectively, and they are real and must not be neglected. The linearization theory assumes that these nonlinear terms are small so that they can be neglected. The fact is that this assumption is not always valid. The reason is explained below. The roll and yaw dynamic systems of an aircraft are harmonic oscillation systems as shown in Math. 3 and Math. 5. As we know for a harmonic system, a resonance phenomenon can be excited by a driving mechanism no matter how small it is as long as its frequency matches the natural frequency of the system. It was found in the inventor's book mentioned above that under certain circumstances the nonlinear terms, (Iy−Iz){dot over (θ)}{dot over (ψ)} and (Ix−Iy){dot over (φ)}{dot over (θ)} can simultaneously excite roll and yaw resonances, respectively. In these cases, the pitch motion becomes unstable and the roll and yaw motions grow exponentially at the same time under the following two conditions, Math. 9 and Math. 10. Such nonlinear instability is a phenomenon of double resonances, i.e. roll resonance in addition to yaw resonance.
                                                        A              p                        >                          A                              P                -                TH                                              =                                    1                              ω                21                                      ⁢                                                                                b                    1                                    ⁢                                      b                    3                                                                                        (                                                                  I                        z                                            -                                              I                        y                                                              )                                    ⁢                                      (                                          (                                                                        I                          y                                                -                                                  I                          x                                                                    )                                                                                            ⁢                                                  ⁢            and                          ⁢                                  ⁢                                            ω              21                        =                                          ω                10                            +                              ω                30                                              ,                                    Math        .                                  ⁢        9                                                                    A              P                        >                          A                              P                -                TH                                              =                                    1                              ω                21                                      ⁢                                                                                b                    1                                    ⁢                                      b                    3                                                                                        (                                                                  I                        z                                            -                                              I                        y                                                              )                                    ⁢                                      (                                                                  I                        y                                            -                                              I                        x                                                              )                                                                        ⁢                                                  ⁢            and                          ⁢                                  ⁢                                            ω              21                        =                                                                          ω                  10                                -                                  ω                  30                                                                            ,                                    Math        .                                  ⁢        10            wherein AP is the pitch response amplitude under the external pitch moment M21 cos(ω21t+α21); ω10=√{square root over (k1/Ix)} and ω30=√{square root over (k3/Iz)} are the roll and yaw natural frequencies, respectively. The nonlinear dynamics says that the pitch motion is stable until the pitch motion reaches the threshold values AP-TH given in Math. 9 or Math. 10. These threshold values show that the vehicle has two dangerous exciting frequencies in pitch. These two frequencies are either the addition of the roll natural frequency ω10 and the yaw natural frequency ω30 or the subtraction of them. At each frequency, the pitch amplitude threshold for pitch to become unstable is inversely proportional to the pitch exciting frequency, proportional to the square root of the product of the roll and yaw damping coefficients, and inversely proportional to the square root of the product of the difference between the yaw and pitch moments of inertia and the difference between the pitch and roll moments of inertia. In summary, there are three factors having effects on the pitch threshold and they are a) the roll and yaw damping, b) the pitch exciting frequency, and c) the distribution of moments of inertia. The most dominant one among these three factors is the damping effect since the damping coefficients could go to zero in certain circumstances, for example, aircraft yaw damper malfunction which makes the yaw damping become zero, or aircraft in stall condition which makes the roll damping become zero. When either the roll damping or the yaw damping is approaching to zero, the pitch threshold is approaching to zero as well and the pitch motion, even it is small but as long as larger than the threshold value, will become unstable and transfer energy to excite roll and yaw resonances. That is the root mechanism behind all these mysterious tragedies mentioned above. In the inventor's book detailed scientific proofs based on analytical, numerical, and experimental results have been given. The inventor's another patent application U.S. Ser. No. 16/153,883 is related to an apparatus used as a demonstrator in the book to demonstrate the phenomenon of nonlinear pitch instability. The inventor also filed another patent application U.S. Ser. No. 16/153,925 for a simulator to simulate the nonlinear dynamics of aircrafts.
The nonlinear instability is always tied with the rotational direction where the moment of inertia is the intermediate between the other two inertias. Depending on the mass distribution of an aircraft, it could have roll, pitch, or yaw nonlinear instability if the roll, pitch, or yaw moment of inertia is the intermediate one, respectively. For example, generally commercial jet aircrafts, like Boeing 737, 747, and A330 etc. will have nonlinear pitch instability problem and military transport aircrafts, like B-52 will have nonlinear roll instability problem.
As shown in Math. 9 and Math. 10, the nonlinear pitch instability thresholds are dependent only on aircraft flight state parameters, for example for pitch instability, like roll and yaw damping coefficients, roll and yaw natural frequencies, and the moments of inertia. Therefore it would be desirable to have a system and method that can calculate actual nonlinear instability threshold and to provide warning signal to pilots based on the real time measured flight parameters.